109 research outputs found
On isolation of singular zeros of multivariate analytic systems
We give a separation bound for an isolated multiple root of a square
multivariate analytic system satisfying that an operator deduced by adding
and a projection of in a direction of the kernel of
is invertible. We prove that the deflation process applied on and this kind
of roots terminates after only one iteration. When is only given
approximately, we give a numerical criterion for isolating a cluster of zeros
of near . We also propose a lower bound of the number of roots in the
cluster.Comment: 17 page
Verified Error Bounds for Isolated Singular Solutions of Polynomial Systems: Case of Breadth One
In this paper we describe how to improve the performance of the
symbolic-numeric method in (Li and Zhi,2009, 2011) for computing the
multiplicity structure and refining approximate isolated singular solutions in
the breadth one case. By introducing a parameterized and deflated system with
smoothing parameters, we generalize the algorithm in (Rump and Graillat, 2009)
to compute verified error bounds such that a slightly perturbed polynomial
system is guaranteed to have a breadth-one multiple root within the computed
bounds.Comment: 20 page
Solving polynomial systems via symbolic-numeric reduction to geometric involutive form
AbstractWe briefly survey several existing methods for solving polynomial systems with inexact coefficients, then introduce our new symbolic-numeric method which is based on the geometric (Jet) theory of partial differential equations. The method is stable and robust. Numerical experiments illustrate the performance of the new method
A Vergleichsstellensatz of Strassen's Type for a Noncommutative Preordered Semialgebra through the Semialgebra of its Fractions
Preordered semialgebras and semirings are two kinds of algebraic structures
occurring in real algebraic geometry frequently and usually play important
roles therein. They have many interesting and promising applications in the
fields of real algebraic geometry, probability theory, theoretical computer
science, quantum information theory, \emph{etc.}. In these applications,
Strassen's Vergleichsstellensatz and its generalized versions, which are
analogs of those Positivstellens\"atze in real algebraic geometry, play
important roles. While these Vergleichsstellens\"atze accept only a commutative
setting (for the semirings in question), we prove in this paper a
noncommutative version of one of the generalized Vergleichsstellens\"atze
proposed by Fritz [\emph{Comm. Algebra}, 49 (2) (2021), pp. 482-499]. The most
crucial step in our proof is to define the semialgebra of the fractions of a
noncommutative semialgebra, which generalizes the definitions in the
literature. Our new Vergleichsstellensatz characterizes the relaxed preorder on
a noncommutative semialgebra induced by all monotone homomorphisms to
by three other equivalent conditions on the semialgebra of its
fractions equipped with the derived preorder, which may result in more
applications in the future.Comment: 28 page
Fourier sum of squares certificates
The non-negativity of a function on a finite abelian group can be certified
by its Fourier sum of squares (FSOS). In this paper, we propose a method of
certifying the non-negativity of an integer-valued function by an FSOS
certificate, which is defined to be an FSOS with a small error. We prove the
existence of exponentially sparse polynomial and rational FSOS certificates and
we provide two methods to validate them. As a consequence of the aforementioned
existence theorems, we propose a semidefinite programming (SDP)-based algorithm
to efficiently compute a sparse FSOS certificate. For applications, we consider
certificate problems for maximum satisfiability (MAX-SAT) and maximum
k-colorable subgraph (MkCS) and demonstrate our theoretical results and
algorithm by numerical experiments
A Counter-Example to the Nichtnegativstellensatz
Klep and Schweighofer asked whether the
Nirgendsnegativsemidefinitheitsstellensatz holds for a symmetric noncommutative
polynomial whose evaluations at bounded self-adjoint operators on any
nontrivial Hilbert space are not negative semidefinite. We provide a
counter-example to this open problem
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